Zero-or-one laws.

roposition

(Kolmogorov zero-or-one law). For an
independent process, each remote event has probability zero or one.

Proof

Let
be a remote event and let
.
We introduce the conditional
probability
Let
.
Then by remoteness of
and independence of
we have
Hence,
Thus
coincides with
on
.
Consequently, see the proposition
(
Random walk space
approximation
),
coincides with
on
.
By construction of
,
We return to the formula
with
:
We set
and
conclude

Proposition

(Preservation of stationary
measure) For a stationary independent process and any
we
have

Proof

The
is the set
moved by one position to the right in the coordinate representation. But by
stationarity and independence the same measure is assigned to all positions.

Proposition

(Hewitt and Savage zero-or-one
law) For a stationary independent process, each permutable set has probability
zero or one.

Proof

Let
be a permutable set. By the proposition
(
Random walk space
approximation
) there is a sequence
such that
.
For every
there is a permutation
such that
.
By
independence,
and by
stationarity,
By
-additivity
of
,
(see the proposition (
Continuity
lemma
))
Hence, we pass the formula
to the limit and
obtain

Proposition

("Infinitely often" zero-or-one
law) Let
be a sequence of subsets of
and
be a random walk. Then the set
is permutable and
is equal to zero or one.

Proof

For any
the r.v.
is invariant with respect to any permutation
.
Hence, the
set
is invariant to such permutation
.
Since the
is decreasing as
then
set
is invariant to any permutation
for any
.
The statement then follows from the proposition
(
Hewitt and Savage zero-or-one
law
).